This does not sound like a relationship that has a strong base. You haven’t felt the need to live together or close by, you prioritise your dog over a life with him, he issues ultimatums over your dog and worse threatens to take you to court for residency (he stand no chance, of course, but could get access).
So based on this I wouldn’t be leaving my friends and family to live with him for the first time when the baby is born,
How do you view this relationship? Was the baby planned?
With someone who starts threats before the baby is even born I would make things as hard as possible.
In fact I would finish the relationship, probably not put him on the BC, definitely not give your baby his surname. Even if the relationship does survive.
Access and knowing their Dad is important if he is committed / interested, but that can happen whether or not he has parental responsibility.
However, he could go to court to establish parental responsibility.